Neural branching heuristics for SAT solving Sebastian Jaszczur, - - PowerPoint PPT Presentation

Neural branching heuristics for SAT solving

Sebastian Jaszczur, Michał Łuszczyk, Henryk Michalewski University of Warsaw AITP 2019, bit.ly/neurheur-aitp2019

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Overview

1. Introduction 2. The neural network 3. Experiments 4. Conclusions

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• 1. Introduction. DPLL algorithm

Our work applies to CDCL, too! DPLL – basic backtracking algorithm for SAT

• solving. For illustration purposes.

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• 2. Models

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Neural network interface

(A v ¬C) ^ (¬B v C) ^ (B v C)

Neural Network 1

A B C

1 1 1 1

¬A ¬B ¬C

CNF formula satisfiable? policy

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bit.ly/neurheur-aitp2019 Invariant TreeNN, LSTM BOW averaging Graph NN “Variable renaming” - invariance No No Yes “Permutation of literals in clause” - invariance No Yes Yes “Permutation of clauses in formula” - invariance No Yes Yes “Negation of all occurrences of variable” - invariance No No Yes

CNF invariants

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CNF formula: graph representation

We can erase the labels and have an equisatisfiable problem!

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Graph neural network for CNF clauses

Inspired by NeuroSAT

Each vertex stores an

• embedding. In each iteration,

each vertex: 1. calculates a message and sends to neighbours, 2. receives the messages and aggregates, 3. calculates its new embedding based on aggregate and previous embedding.

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Attention mechanism in aggregation

Receiver Sender Senders Sender V vectors Sender K vectors Q vector Sender Message logit-weights Sender Message weights Aggregated message

Dot product Multiply MLP MLP MLP Sigmoid NeuroSAT: sum,

• ur model: attention with sigmoid

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Dataset SR(x)

• x - number of variables
• CNF distribution introduced in NeuroSAT
• formulas difficult for SAT solvers
• labels: generated with Glucose

Average number of clauses Average formula size Average time for MiniSAT SR(30): 300±33 1480±175 0.007 SR(110) 1060±50 5100±287 0.137 SR(150) 1450±60 6930±320 3.406 Problem difficulty:

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• 2. Trained models.

We trained 3-5 models on each of the following datasets. Mean and stdev over 3-5 trained models.

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• 3. Experiments

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Performance with DPLL compared to static heuristics; at step limit 1000

Learned heuristic better than DLIS. JW-OS better at SR(50)-SR(70) and worse at SR(90)-SR(110).

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Hybrid vs JW-OS guiding DPLL and CDCL

DPLL CDCL

Hybrid - use Model SR(50) if sat probability > 0.3, otherwise fallbacks to JW-OS. Why? 1. Faster 2. NN policy isn’t trained on unsatisfiable formulas.

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Attention experiment

y axis: policy error, average and std over 3-5 runs

• Attention

significantly better,

• except for SR(30)

l20 and SR(50) l40.

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Our contributions

• Learned branching

heuristic using a GNN

• Modifying the

NeuroSAT-inspired network with attention

Our workshop paper: bit.ly/neurheur-iclr2019 Future work

• Optimise for the number of steps.
• Curriculum learning.
• Integrate with restart policy and

clause learning and forgetting decision.

• Compare to VSIDS and other

dynamic heuristics.

• Unsat certificates.

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(e|~f|d|a|~b|~j)&(~e|g|b)&(f|j|g)&(e|~f|j|~i)&(~j|f|~d)&(~c|~f|~h)&(~a|~j|~d)&(~j|h|~b| ~f)&(~h|~c|i|~b|j|e)&(~c|~e|a|b|i)&(~f|b|g|~i)&(~a|i|h|f|c)&(~d|f|g|~c|a)&(a|d|~c|~h)&( c|~a|~h|d)&(f|~i|~a)&(d|b|f|~g|a)&(e|d|c|~f|~j)&(~f|d|~a)&(~c|~g|~d|~f|j)&(f|e|j)&(d|c| ~a|f|~j|e)&(i|~c|d|~j|h|b|a|~g|~f)&(j|i|~d)&(e|f|d)&(f|g|~a)&(~b|h|d)&(j|f|~b|g)&(~e|~g| ~h)&(e|h|~c|~b|d|a|g)&(~e|~j|~i|d|f)&(h|e|~g|d)&(~f|h|c|g|~j|b|d|~i|~a)&(~d|~j|~h|i|~c )&(~a|~c|~d)&(d|~a|f)&(a|h|d)&(~b|~g|~f|~i|a|d)&(~a|~f|~g|j)&(e|~b|j|f)&(i|~g|~f|~e)& (a|b|~h|~j|~f|~i)&(a|j|c)&(g|j|~c|~f)&(~e|d|~f|~c)&(j|~c|f)&(e|i|~f|j|b|g)&(~d|a|c|e)&(~b |~a|~h)&(a|~j|h|e|i|c)&(~d|~a|g|h|~c|~f|j)&(~h|~e|~d)&(h|~j|~b|~f|~e|~g|d)&(~i|~j|a|d )&(~i|~h|~f|~b|c)&(~f|e|a)&(j|b|~i)&(~c|j|~i|a|~h|e|d)&(~i|~h|c|~e|j|~d|~g)&(b|~c|j|i)&( e|~d|~a|~g|~h)&(c|~b|i|~h)&(~j|h|c)&(~c|i|g)&(f|a|~i)&(~e|h|~j|~c|~d)&(a|b|~f)&(j|~h| i|d)&(~d|a|e)&(h|~j|d)&(d|~f|~h|~e|~b)&(~b|~f|j|~c|h|~i)&(~b|f|~j|d|~g)&(~a|i|~b|~c)& (d|e|~c|j|~h)&(~b|~d|~i|~j|~a)&(f|j|i|h|~g|d|~a|~c)&(g|f|~e)&(~e|~h|d|j)&(i|j|~a|g|~e|~ h)&(~e|~g|j)&(c|~h|e|~j|~d)&(i|~f|~j)&(b|~h|~g|j|f|i|e|~d)&(~j|e|~i|~a|d|~g)

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Why sigmoidal attention?

• Attention with sigmoid resembles aggregation with sum while attention with

softmax resembles aggregation with average,

• average loses important information e.g. it cannot count the neighbours.

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Why we compare steps rather than time

• Time optimisations are possible (parallel execution, simpler model,

non-Python implementation), it’s engineering work,

• bigger models determines the upper bound,
• we expect better time at sufficiently large instances:

○ actually, we’re better than some heuristics now.

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How NeuroSAT works x axis - iteration number, sat probability at each literal node

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Usage

• We take a formula, and we predict SAT probability and policy probability
• SAT probability, computed for whole formula:

○ Ground truth is "is whole formula satisfiable?" - like NeuroSAT ○ We do linear regression on every node's embedding. We output sigmoid of sum of the results

• f those linear regression.
• Policy probability, computed separately for each literal:

○ Ground truth is "is there a solution to this formula with this literal?". ○ We do logistic regression on every literal node's embedding.

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